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   <h4 class="subsectionHead"><span class="titlemark">2.4   </span> <a 
 id="x9-140002.4"></a>Plane Stress Elements</h4>
<!--l. 372--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.4.1   </span> <a 
 id="x9-150002.4.1"></a>PlaneStress2d</h5>
<!--l. 373--><p class="noindent" >Represents isoparametric four-node quadrilateral plane-stress finite element. Each node has 2 degrees of freedom.
Structure should be defined in x,y plane. The nodes should be numbered anti-clockwise (positive rotation around
z-axis). The element features are summarized in Table&#x00A0;<a 
href="#x9-150027">7<!--tex4ht:ref: planestress2dsummary --></a>.
<!--l. 379--><p class="indent" >   The generalization of this element, that can be positioned arbitrarily in space is <span 
class="cmtt-10">linquad3dplanestress</span>
element. This element requires 3 displacement degrees of freedon in each node and assumes, that the
element geometry is flat, i.e. all nodes are in the same plane. The element features are summarized in
Table&#x00A0;<a 
href="#x9-150038">8<!--tex4ht:ref: linquad3dplanestresssummary --></a>.
<!--l. 383--><p class="indent" >   <hr class="figure"><div class="figure" 
>
                                                                                           
                                                                                           
<a 
 id="x9-150015"></a>
                                                                                           
                                                                                           
<!--l. 1--><p class="noindent" ><img 
src="elementlibmanual-figure4.png" alt="PIC"  
>
<br /> <div class="caption" 
><span class="id">Figure&#x00A0;5: </span><span  
class="content">PlaneStress2d element. Node numbering, edge numbering and definition of local edge c.s.(a).</span></div><!--tex4ht:label?: x9-150015 -->
                                                                                           
                                                                                           
<!--l. 390--><p class="indent" >   </div><hr class="endfigure">
<div class="center" 
>
<!--l. 392--><p class="noindent" >
<div class="table">
<!--l. 392--><p class="noindent" ><a 
 id="x9-150027"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-8" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-8-1g"><col 
id="TBL-8-1"></colgroup><colgroup id="TBL-8-2g"><col 
id="TBL-8-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-8-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-8-1-2"  
class="td11"><!--l. 392--><p class="noindent" ><span 
class="cmbx-10">planestress2d</span>                                                 </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-8-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-8-2-2"  
class="td11"><!--l. 392--><p class="noindent" >2D quadrilateral element for plane stress analysis            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-8-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-8-3-2"  
class="td11"><!--l. 392--><p class="noindent" >[<span 
class="cmtt-10">NIP</span>&#x00A0;#<span 
class="cmr-5">(in)</span>]                                                               </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-8-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-4-1"  
class="td11">Parameters           </td><td  style="white-space:wrap; text-align:left;" id="TBL-8-4-2"  
class="td11"><!--l. 393--><p class="noindent" ><span 
class="cmtt-10">NIP</span>: allows to set the number of integration points          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-8-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-5-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-8-5-2"  
class="td11"><!--l. 394--><p class="noindent" >Two dofs (u-displacement, v-displacement) are required in
each node.                                                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-8-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-6-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-8-6-2"  
class="td11"><!--l. 395--><p class="noindent" >Linear approximation of displacements and geometry.      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-8-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-7-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-8-7-2"  
class="td11"><!--l. 398--><p class="noindent" >Integration   of   membrane   strain   terms   using   Gauss
integration formula in 1, 4 (default), 9 or 16 integration
points. The default number of integration points used can
be overloaded using <span 
class="cmtt-10">NIP </span>parameter. Reduced integration for
shear terms is employed. Shear terms are always integrated
using the 1-point integration rule.                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-8-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-8-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-8-8-2"  
class="td11"><!--l. 399--><p class="noindent" >Nonlocal  constitutive  support,  Geometric  nonlinearity
support.                                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-8-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-9-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-8-9-2"  
class="td11"><!--l. 400--><p class="noindent" >cross section thickness is required.                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-8-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-10-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-8-10-2"  
class="td11"><!--l. 403--><p class="noindent" >Body loads are supported. Boundary loads are supported
and  computed  using  numerical  integration.  The  side
numbering  is  following.  Each  i-th  element  side  begins
in  i-th  element  node  and  ends  on  next  element  node
(i+1-th  node  or  1-st  node,  in  the  case  of  side  number
4).  The  local  positive  edge  x-axis  coincides  with  side
direction, the positive local edge y-axis is rotated 90 degrees
anti-clockwise (see fig. (<a 
href="#x9-150015">5<!--tex4ht:ref: Planestress2dfig --></a>)).                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-8-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-11-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-8-11-2"  
class="td11"><!--l. 404--><p class="noindent" >0, 1.                                                                       </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-8-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-8-12-2"  
class="td11"><!--l. 405--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-8-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-13-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-8-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-8-14-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;7: </span><span  
class="content">planestress2d element summary</span></div><!--tex4ht:label?: x9-150027 -->
</div><hr class="endfloat" />
</div>
</div>
<div class="center" 
>
<!--l. 408--><p class="noindent" >
<div class="table">
                                                                                           
                                                                                           
<!--l. 408--><p class="noindent" ><a 
 id="x9-150038"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-9" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-9-1g"><col 
id="TBL-9-1"></colgroup><colgroup id="TBL-9-2g"><col 
id="TBL-9-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-9-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-9-1-2"  
class="td11"><!--l. 408--><p class="noindent" ><span 
class="cmbx-10">linquad3dplanestress</span>                                        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-9-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-9-2-2"  
class="td11"><!--l. 408--><p class="noindent" >3D quadrilateral element for plane stress analysis            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-9-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-9-3-2"  
class="td11"><!--l. 408--><p class="noindent" >[<span 
class="cmtt-10">NIP</span>&#x00A0;#<span 
class="cmr-5">(in)</span>]                                                               </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-9-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-4-1"  
class="td11">Parameters           </td><td  style="white-space:wrap; text-align:left;" id="TBL-9-4-2"  
class="td11"><!--l. 409--><p class="noindent" ><span 
class="cmtt-10">NIP</span>: allows to set the number of integration points          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-9-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-5-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-9-5-2"  
class="td11"><!--l. 410--><p class="noindent" >Three
dofs (u-displacement, v-displacement, w-displacement) are
required in each node.                                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-9-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-6-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-9-6-2"  
class="td11"><!--l. 411--><p class="noindent" >Linear approximation of displacements and geometry.      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-9-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-7-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-9-7-2"  
class="td11"><!--l. 413--><p class="noindent" >Integration   of   membrane   strain   terms   using   Gauss
integration formula in 1, 4 (default), 9 or 16 integration
points. The default number of integration points used can
be overloaded using <span 
class="cmtt-10">NIP </span>parameter. Reduced integration for
shear terms is employed. Shear terms are always integrated
using the 1-point integration rule.                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-9-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-8-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-9-8-2"  
class="td11"><!--l. 414--><p class="noindent" >Nonlocal  constitutive  support,  Geometric  nonlinearity
support.                                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-9-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-9-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-9-9-2"  
class="td11"><!--l. 415--><p class="noindent" >cross section thickness is required.                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-9-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-10-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-9-10-2"  
class="td11"><!--l. 418--><p class="noindent" >Body loads are supported. Boundary loads are supported
and  computed  using  numerical  integration.  The  side
numbering  is  following.  Each  i-th  element  side  begins
in  i-th  element  node  and  ends  on  next  element  node
(i+1-th  node  or  1-st  node,  in  the  case  of  side  number
4).  The  local  positive  edge  x-axis  coincides  with  side
direction, the positive local edge y-axis is rotated 90 degrees
anti-clockwise (see fig. (<a 
href="#x9-150015">5<!--tex4ht:ref: Planestress2dfig --></a>)).                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-9-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-11-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-9-11-2"  
class="td11"><!--l. 419--><p class="noindent" >0, 1.                                                                       </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-9-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-9-12-2"  
class="td11"><!--l. 420--><p class="noindent" >Basic  functionality  tested,  element  loads  need  further
testing.                                                                   </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-9-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-13-1"  
class="td11">                </td></tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-9-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-9-14-1"  
class="td11"></td></tr></table>
</div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;8: </span><span  
class="content">linquad3dplanestress element summary</span></div><!--tex4ht:label?: x9-150038 -->
</div><hr class="endfloat" />
</div>
</div>
   <h5 class="subsubsectionHead"><span class="titlemark">2.4.2   </span> <a 
 id="x9-160002.4.2"></a>QPlaneStress2d</h5>
<!--l. 425--><p class="noindent" >Implementation of quadratic isoparametric eight-node quadrilateral plane-stress finite element. Each node has 2
degrees of freedom. The node numbering is anti-clockwise and is explained in fig. (<a 
href="#x9-160016">6<!--tex4ht:ref: qplanstrssfig --></a>). The element features are
summarized in Table&#x00A0;<a 
href="#x9-160029">9<!--tex4ht:ref: qplanestress2dsummary --></a>.
<!--l. 430--><p class="indent" >   <hr class="figure"><div class="figure" 
>
                                                                                           
                                                                                           
<a 
 id="x9-160016"></a>
                                                                                           
                                                                                           
<!--l. 1--><p class="noindent" ><img 
src="elementlibmanual-figure5.png" alt="PIC"  
>
<br /> <div class="caption" 
><span class="id">Figure&#x00A0;6: </span><span  
class="content">QPlaneStress2d element - node numbering.</span></div><!--tex4ht:label?: x9-160016 -->
                                                                                           
                                                                                           
<!--l. 437--><p class="indent" >   </div><hr class="endfigure">
<div class="center" 
>
<!--l. 439--><p class="noindent" >
<div class="table">
<!--l. 439--><p class="noindent" ><a 
 id="x9-160029"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-10" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-10-1g"><col 
id="TBL-10-1"></colgroup><colgroup id="TBL-10-2g"><col 
id="TBL-10-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-10-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-10-1-2"  
class="td11"><!--l. 439--><p class="noindent" ><span 
class="cmbx-10">qplanestress2d</span>                                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-10-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-10-2-2"  
class="td11"><!--l. 439--><p class="noindent" >2D quadratic isoparametric plane stress element              </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-10-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-10-3-2"  
class="td11"><!--l. 439--><p class="noindent" >[<span 
class="cmtt-10">NIP</span>&#x00A0;#<span 
class="cmr-5">(in)</span>]                                                               </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-10-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-4-1"  
class="td11">Parameters           </td><td  style="white-space:wrap; text-align:left;" id="TBL-10-4-2"  
class="td11"><!--l. 440--><p class="noindent" ><span 
class="cmtt-10">NIP</span>: allows to set the number of integration points          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-10-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-5-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-10-5-2"  
class="td11"><!--l. 441--><p class="noindent" >Two dofs (u-displacement, v-displacement) are required in
each node.                                                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-10-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-6-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-10-6-2"  
class="td11"><!--l. 442--><p class="noindent" >Quadratic approximation of displacements and geometry.  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-10-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-7-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-10-7-2"  
class="td11"><!--l. 443--><p class="noindent" >Full integration using Gauss integration formula in 4 (the
default), 9 or 16 integration points. The default number
of  integration  points  used  can  be  overloaded  using  <span 
class="cmtt-10">NIP</span>
parameter.                                                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-10-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-8-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-10-8-2"  
class="td11"><!--l. 444--><p class="noindent" >Adaptivity support.                                                   </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-10-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-9-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-10-9-2"  
class="td11"><!--l. 445--><p class="noindent" >Cross section thickness is required.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-10-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-10-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-10-10-2"  
class="td11"><!--l. 446--><p class="noindent" >Body and boundary loads are supported.                       </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-10-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-11-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-10-11-2"  
class="td11"><!--l. 447--><p class="noindent" >0, 1.                                                                       </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-10-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-10-12-2"  
class="td11"><!--l. 448--><p class="noindent" >Stable                                                                     </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-10-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-13-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-10-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-10-14-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;9: </span><span  
class="content">qplanestress2d element summary</span></div><!--tex4ht:label?: x9-160029 -->
</div><hr class="endfloat" />
</div>
</div>
   <h5 class="subsubsectionHead"><span class="titlemark">2.4.3   </span> <a 
 id="x9-170002.4.3"></a>TrPlaneStress2d</h5>
<!--l. 455--><p class="noindent" >Implements an triangular three-node constant strain plane-stress finite element. Each node has 2 degrees of freedom.
The node numbering is anti-clockwise. The element features are summarized in Table&#x00A0;<a 
href="#x9-1700210">10<!--tex4ht:ref: trplanestress2dsummary --></a>.
<!--l. 459--><p class="indent" >   <hr class="figure"><div class="figure" 
>
                                                                                           
                                                                                           
<a 
 id="x9-170017"></a>
                                                                                           
                                                                                           
<!--l. 1--><p class="noindent" ><img 
src="elementlibmanual-figure6.png" alt="PIC"  
>
<br /> <div class="caption" 
><span class="id">Figure&#x00A0;7: </span><span  
class="content">TrPlaneStress2d element - node and side numbering.</span></div><!--tex4ht:label?: x9-170017 -->
                                                                                           
                                                                                           
<!--l. 466--><p class="indent" >   </div><hr class="endfigure">
<div class="center" 
>
<!--l. 468--><p class="noindent" >
<div class="table">
<!--l. 468--><p class="noindent" ><a 
 id="x9-1700210"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-11" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-11-1g"><col 
id="TBL-11-1"></colgroup><colgroup id="TBL-11-2g"><col 
id="TBL-11-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-11-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-11-1-2"  
class="td11"><!--l. 468--><p class="noindent" ><span 
class="cmbx-10">trplanestress2d</span>                                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-11-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-11-2-2"  
class="td11"><!--l. 468--><p class="noindent" >2D linear triangular isoparametric plane stress element     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-11-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-11-3-2"  
class="td11"><!--l. 468--><p class="noindent" >-                                                                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-11-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-11-4-2"  
class="td11"><!--l. 469--><p class="noindent" >Two dofs (u-displacement, v-displacement) are required in
each node.                                                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-11-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-11-5-2"  
class="td11"><!--l. 470--><p class="noindent" >Linear approximation of displacements and geometry.      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-11-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-11-6-2"  
class="td11"><!--l. 471--><p class="noindent" >Integration of membrane strain terms using one point gauss
integration formula.                                                   </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-11-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-7-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-11-7-2"  
class="td11"><!--l. 472--><p class="noindent" >Nonlocal   constitutive   support,   Edge   load   support,
Geometric nonlinearity support, Adaptivity support.        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-11-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-8-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-11-8-2"  
class="td11"><!--l. 473--><p class="noindent" >Cross section thickness is required.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-11-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-9-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-11-9-2"  
class="td11"><!--l. 480--><p class="noindent" >Body loads are supported. Boundary loads are supported
and are computed using numerical integration. The side
numbering  is  following.  Each  i-th  element  side  begins
in  i-th  element  node  and  ends  on  next  element  node
(i+1-th  node  or  1-st  node,  in  the  case  of  side  number
3).  The  local  positive  edge  x-axis  coincides  with  side
direction, the positive local edge y-axis is rotated 90 degrees
anti-clockwise (see fig. (<a 
href="#x9-170017">7<!--tex4ht:ref: TrPlanestressfig --></a>)).                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-11-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-10-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-11-10-2"  
class="td11"><!--l. 481--><p class="noindent" >0, 1.                                                                       </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-11-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-11-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-11-11-2"  
class="td11"><!--l. 482--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-11-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-12-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-11-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-11-13-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;10: </span><span  
class="content">trplanestress2d element summary</span></div><!--tex4ht:label?: x9-1700210 -->
</div><hr class="endfloat" />
</div>
</div>
   <h5 class="subsubsectionHead"><span class="titlemark">2.4.4   </span> <a 
 id="x9-180002.4.4"></a>QTrPlStr</h5>
<!--l. 488--><p class="noindent" >Implementation of quadratic six-node plane-stress finite element. Each node has 2 degrees of freedom.
Node numbering is anti-clockwise and is shown in fig. (<a 
href="#x9-180018">8<!--tex4ht:ref: qtrplanstressfig --></a>). The element features are summarized in
Table&#x00A0;<a 
href="#x9-1800211">11<!--tex4ht:ref: qtrplstrsummary --></a>.
<!--l. 492--><p class="indent" >   <hr class="figure"><div class="figure" 
>
                                                                                           
                                                                                           
<a 
 id="x9-180018"></a>
                                                                                           
                                                                                           
<!--l. 2--><p class="noindent" ><img 
src="elementlibmanual-figure7.png" alt="PIC"  
>
<br /> <div class="caption" 
><span class="id">Figure&#x00A0;8: </span><span  
class="content">QTrPlStr element - node and side numbering.</span></div><!--tex4ht:label?: x9-180018 -->
                                                                                           
                                                                                           
<!--l. 499--><p class="indent" >   </div><hr class="endfigure">
<div class="center" 
>
<!--l. 501--><p class="noindent" >
<div class="table">
<!--l. 501--><p class="noindent" ><a 
 id="x9-1800211"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-12" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-12-1g"><col 
id="TBL-12-1"></colgroup><colgroup id="TBL-12-2g"><col 
id="TBL-12-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-12-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-12-1-2"  
class="td11"><!--l. 501--><p class="noindent" ><span 
class="cmbx-10">qtrplstr</span>                                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-12-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-12-2-2"  
class="td11"><!--l. 501--><p class="noindent" >2D quadratic triangular plane stress element                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-12-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-12-3-2"  
class="td11"><!--l. 501--><p class="noindent" >[<span 
class="cmtt-10">NIP</span>&#x00A0;#<span 
class="cmr-5">(in)</span>]                                                               </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-12-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-4-1"  
class="td11">Parameters           </td><td  style="white-space:wrap; text-align:left;" id="TBL-12-4-2"  
class="td11"><!--l. 502--><p class="noindent" ><span 
class="cmtt-10">NIP</span>: allows to set the number of integration points          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-12-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-5-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-12-5-2"  
class="td11"><!--l. 503--><p class="noindent" >Two dofs (u-displacement, v-displacement) are required in
each node.                                                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-12-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-6-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-12-6-2"  
class="td11"><!--l. 504--><p class="noindent" >Quadratic approximation of displacements and geometry.  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-12-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-7-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-12-7-2"  
class="td11"><!--l. 506--><p class="noindent" >Full integration using gauss integration formula in 4 points
(the default) or in 7 points (using <span 
class="cmtt-10">NIP </span>parameter).          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-12-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-8-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-12-8-2"  
class="td11"><!--l. 507--><p class="noindent" >Adaptivity support (error indicator).                            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-12-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-9-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-12-9-2"  
class="td11"><!--l. 508--><p class="noindent" >Cross section thickness is required.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-12-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-10-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-12-10-2"  
class="td11"><!--l. 509--><p class="noindent" >Boundary loads are supported.                                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-12-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-11-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-12-11-2"  
class="td11"><!--l. 510--><p class="noindent" >0, 1.                                                                       </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-12-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-12-12-2"  
class="td11"><!--l. 511--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-12-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-13-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-12-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-12-14-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;11: </span><span  
class="content">qtrplstr element summary</span></div><!--tex4ht:label?: x9-1800211 -->
</div><hr class="endfloat" />
</div>
</div>
   <h5 class="subsubsectionHead"><span class="titlemark">2.4.5   </span> <a 
 id="x9-190002.4.5"></a>TrPlaneStrRot</h5>
<!--l. 518--><p class="noindent" >Implementation of triangular three-node plane-stress finite element with independent rotation field. Each node has 3
degrees of freedom. The element features are summarized in Table&#x00A0;<a 
href="#x9-1900112">12<!--tex4ht:ref: trplanestrrotsummary --></a>.
<!--l. 522--><p class="indent" >   The generalization of this element, that can be positioned arbitrarily in space is <span 
class="cmtt-10">trplanestrrot3d</span>
element. This element requires 6 degrees of freedon in each node. The element features are summarized in
Table&#x00A0;<a 
href="#x9-1900213">13<!--tex4ht:ref: trplanestrrot3dsummary --></a>.
<!--l. 525--><p class="indent" >   The implementation is based on the following paper: Ibrahimbegovic, A., Taylor, R.L., Wilson, E. L.: A robust
membrane qudritelar element with rotational degrees of freedom, Int. J. Num. Meth. Engng., 30, 445-457, 1990. The
rotation field is defined as <span 
class="cmmi-10">&omega; </span>= <img 
src="elementlibmanual0x.png" alt="1
2"  class="frac" align="middle">(<img 
src="elementlibmanual1x.png" alt="dv
dx"  class="frac" align="middle"> <span 
class="cmsy-10">&minus;</span><img 
src="elementlibmanual2x.png" alt="du-
dy"  class="frac" align="middle">) = <span 
class="cmsy-10">&nabla;</span><sub><span 
class="cmmi-7">u</span></sub><span 
class="cmmib-10">u</span>. The following form of potential energy functial is
assumed:
   <table 
class="equation-star"><tr><td>
   <center class="math-display" >
<img 
src="elementlibmanual3x.png" alt="    1 &int;   T      &int;   T               &int;   T
&Pi; = 2  &Omega;&sigma;  &#x03B5; d&Omega;+  &Omega; &tau; (&nabla;uu &minus; &omega;) d&Omega; &minus;  &Omega;X  u d&Omega;
" class="math-display" ></center></td></tr></table>
<!--l. 529--><p class="nopar" >
where <span 
class="cmmib-10">&tau;</span> is pseudo-stress (component of anti-symmetric stress tensor) working on dislocation (<span 
class="cmsy-10">&nabla;</span><sub><span 
class="cmmi-7">u</span></sub><span 
class="cmmib-10">u</span> <span 
class="cmsy-10">&minus; </span><span 
class="cmmi-10">&omega;</span>);
the following constitutive relation foris assumed: <span 
class="cmmib-10">&tau;</span> = <span 
class="cmmi-10">G</span>(<span 
class="cmsy-10">&nabla;</span><sub><span 
class="cmmi-7">u</span></sub><span 
class="cmmib-10">u</span> <span 
class="cmsy-10">&minus; </span><span 
class="cmmi-10">&omega;</span>), where <span 
class="cmmi-10">G </span>is elasticity modulus in
shear.
<div class="center" 
>
<!--l. 532--><p class="noindent" >
                                                                                           
                                                                                           
<div class="table">
<!--l. 532--><p class="noindent" ><a 
 id="x9-1900112"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-13" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-13-1g"><col 
id="TBL-13-1"></colgroup><colgroup id="TBL-13-2g"><col 
id="TBL-13-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-13-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-13-1-2"  
class="td11"><!--l. 532--><p class="noindent" ><span 
class="cmbx-10">trplanestrrot</span>                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-13-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-13-2-2"  
class="td11"><!--l. 532--><p class="noindent" >2D linear triangular plane stress element with rotational
DOFs                                                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-13-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-13-3-2"  
class="td11"><!--l. 532--><p class="noindent" >[<span 
class="cmtt-10">NIP</span>&#x00A0;#<span 
class="cmr-5">(in)</span>] [<span 
class="cmtt-10">NIPRot</span>&#x00A0;#<span 
class="cmr-5">(in)</span>]                                           </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-13-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-4-1"  
class="td11">Parameters           </td><td  style="white-space:wrap; text-align:left;" id="TBL-13-4-2"  
class="td11"><!--l. 533--><p class="noindent" ><span 
class="cmtt-10">NIP</span>:  allows  to  set  the  number  of  integration  points  for
integration of membrane terms.                                   </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-13-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-5-1"  
class="td11">                </td><td  style="white-space:wrap; text-align:left;" id="TBL-13-5-2"  
class="td11"><!--l. 534--><p class="noindent" ><span 
class="cmtt-10">NIPRot</span>: allows to set the number of integration points for
integration of terms associated to rotational field.            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-13-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-6-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-13-6-2"  
class="td11"><!--l. 535--><p class="noindent" >Three dofs (u-displacement, v-displacement, z-rotation) are
required in each node.                                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-13-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-7-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-13-7-2"  
class="td11"><!--l. 536--><p class="noindent" >Linear approximation of displacements and geometry.      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-13-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-8-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-13-8-2"  
class="td11"><!--l. 538--><p class="noindent" >Integration   of   membrane   strain   terms   using   gauss
integration  formula  in  4  points  (default)  or  using  1  or
7  points  (using  <span 
class="cmtt-10">NIP </span>parameter).  Integration  of  strains
associated with rotational field integration using 1 point is
default (4 and 7 points rules can be specified using <span 
class="cmtt-10">NIPRot</span>
parameter).                                                             </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-13-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-9-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-13-9-2"  
class="td11"><!--l. 539--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-13-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-10-1"  
class="td11">CS properties </td> <td  style="white-space:wrap; text-align:left;" id="TBL-13-10-2"  
class="td11"><!--l. 540--><p class="noindent" >Cross section thickness is required.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-13-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-11-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-13-11-2"  
class="td11"><!--l. 541--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-13-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-12-1"  
class="td11">Nlgeo </td> <td  style="white-space:wrap; text-align:left;" id="TBL-13-12-2"  
class="td11"><!--l. 542--><p class="noindent" >0.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-13-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-13-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-13-13-2"  
class="td11"><!--l. 543--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-13-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-14-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-13-15-"><td  style="white-space:nowrap; text-align:left;" id="TBL-13-15-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;12: </span><span  
class="content">trplanestrrot element summary</span></div><!--tex4ht:label?: x9-1900112 -->
</div><hr class="endfloat" />
</div>
</div>
<div class="center" 
>
<!--l. 546--><p class="noindent" >
<div class="table">
<!--l. 546--><p class="noindent" ><a 
 id="x9-1900213"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-14" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-14-1g"><col 
id="TBL-14-1"></colgroup><colgroup id="TBL-14-2g"><col 
id="TBL-14-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-14-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-14-1-2"  
class="td11"><!--l. 546--><p class="noindent" ><span 
class="cmbx-10">trplanestrrot3d</span>                                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-14-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-14-2-2"  
class="td11"><!--l. 546--><p class="noindent" >3D linear triangular plane stress element with rotational
DOFs                                                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-14-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-14-3-2"  
class="td11"><!--l. 546--><p class="noindent" >[<span 
class="cmtt-10">NIP</span>&#x00A0;#<span 
class="cmr-5">(in)</span>] [<span 
class="cmtt-10">NIPRot</span>&#x00A0;#<span 
class="cmr-5">(in)</span>]                                           </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-14-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-4-1"  
class="td11">Parameters           </td><td  style="white-space:wrap; text-align:left;" id="TBL-14-4-2"  
class="td11"><!--l. 547--><p class="noindent" ><span 
class="cmtt-10">NIP</span>:  allows  to  set  the  number  of  integration  points  for
integration of membrane terms.                                   </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-14-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-5-1"  
class="td11">                </td><td  style="white-space:wrap; text-align:left;" id="TBL-14-5-2"  
class="td11"><!--l. 548--><p class="noindent" ><span 
class="cmtt-10">NIPRot</span>: allows to set the number of integration points for
integration of terms associated to rotational field.            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-14-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-6-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-14-6-2"  
class="td11"><!--l. 549--><p class="noindent" >Six dofs (u-displacement, v-displacement, w-displacement,
x-rotation, y-rotation, z-rotation) are required in each node.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-14-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-7-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-14-7-2"  
class="td11"><!--l. 550--><p class="noindent" >Linear approximation of displacements and geometry.      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-14-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-8-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-14-8-2"  
class="td11"><!--l. 552--><p class="noindent" >Integration   of   membrane   strain   terms   using   gauss
integration  formula  in  4  points  (default)  or  using  1  or
7  points  (using  <span 
class="cmtt-10">NIP </span>parameter).  Integration  of  strains
associated with rotational field integration using 1 point is
default (4 and 7 points rules can be specified using <span 
class="cmtt-10">NIPRot</span>
parameter).                                                             </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-14-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-9-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-14-9-2"  
class="td11"><!--l. 553--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-14-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-10-1"  
class="td11">CS properties </td> <td  style="white-space:wrap; text-align:left;" id="TBL-14-10-2"  
class="td11"><!--l. 554--><p class="noindent" >Cross section thickness is required.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-14-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-11-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-14-11-2"  
class="td11"><!--l. 555--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-14-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-12-1"  
class="td11">Nlgeo </td> <td  style="white-space:wrap; text-align:left;" id="TBL-14-12-2"  
class="td11"><!--l. 556--><p class="noindent" >0.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-14-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-13-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-14-13-2"  
class="td11"><!--l. 557--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-14-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-14-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-14-15-"><td  style="white-space:nowrap; text-align:left;" id="TBL-14-15-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;13: </span><span  
class="content">trplanestrrot3d element summary</span></div><!--tex4ht:label?: x9-1900213 -->
</div><hr class="endfloat" />
</div>
</div>
                                                                                           
                                                                                           
<!--l. 560--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.4.6   </span> <a 
 id="x9-200002.4.6"></a>TrPlaneStressRotAllman</h5>
<!--l. 561--><p class="noindent" >Implementation of triangular three-node plane-stress with nodal rotations. Each node has 3 degrees of freedom. The
element features are summarized in Table&#x00A0;<a 
href="#x9-2000314">14<!--tex4ht:ref: TrPlaneStressRotAllmansummary --></a>.
<!--l. 565--><p class="indent" >   The generalization of this element, that can be positioned arbitrarily in space is <span 
class="cmtt-10">trplanestressrotallman3d</span>
element. This element requires 6 degrees of freedon in each node. The element features are summarized in
Table&#x00A0;<a 
href="#x9-2000415">15<!--tex4ht:ref: trplanestressrotallman3dsummary --></a>.
<!--l. 568--><p class="indent" >   The implementation is based on the following paper: Allman, D.J.: A compatible triangular element including
vertex rotations for plane elasticity analysis, Computers &amp; Structures, vol. 19, no. 1-2, pp. 1-8, 1984. The element is
based on plane stress element with quadratic interpolation. The displacements in midside nodes are expressed using
vertex displacements and vertex rotations (for edge normal displacement component); the tangential
component is interpolated from vertex values. For particular element side starting at i-th vertex and
ending in j-th vertex the normal and tangential displacements at edge midpoint can be expressed as
<div class="eqnarray">
   <center class="math-display" >
<img 
src="elementlibmanual4x.png" alt="           uni + unj  l
un |l&#x2215;2  =   ---2----+ 8 (&omega;i &minus; &omega;j)
           uti + utj
 ut|l&#x2215;2  =   ---2----
" class="math-display" ></center>
</div>where <span 
class="cmmi-10">l </span>is edge length. This allows to express global displacements in element midside nodes using vertex
displacements and rotations. For a single edge, one obtains: <div class="eqnarray">
   <center class="math-display" >
<img 
src="elementlibmanual5x.png" alt="u|    =  &minus; uni +-unj+-l(&omega;  &minus; &omega; )&Delta;yji + (ut1-+-ut2)&Delta;xji
  l&#x2215;2          2      8  i    j  l        2      l
v|    =  uni +-unj + l(&omega; &minus; &omega; )&Delta;xji-+(ut1 +-ut2)&Delta;yji
  l&#x2215;2         2      8  i   j   l        2      l
" class="math-display" ></center>
</div>
<div class="center" 
>
<!--l. 582--><p class="noindent" >
<div class="table">
                                                                                           
                                                                                           
<!--l. 582--><p class="noindent" ><a 
 id="x9-2000314"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-15" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-15-1g"><col 
id="TBL-15-1"></colgroup><colgroup id="TBL-15-2g"><col 
id="TBL-15-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-15-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-15-1-2"  
class="td11"><!--l. 582--><p class="noindent" ><span 
class="cmbx-10">trplanestressrotallman</span>                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-15-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-15-2-2"  
class="td11"><!--l. 582--><p class="noindent" >2D linear triangular plane stress element with rotational
DOFs                                                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-15-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-15-3-2"  
class="td11"><!--l. 582--><p class="noindent" >                                                 </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-15-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-15-4-2"  
class="td11"><!--l. 583--><p class="noindent" >Three dofs (u-displacement, v-displacement, z-rotation) are
required in each node.                                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-15-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-15-5-2"  
class="td11"><!--l. 584--><p class="noindent" >Linear approximation of geometry, quadratic interpolation
of displacements.                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-15-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-15-6-2"  
class="td11"><!--l. 585--><p class="noindent" >Integration   of   membrane   strain   terms   using   gauss
integration formula in 4 points.                                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-15-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-7-1"  
class="td11">Zero energy mode  </td><td  style="white-space:wrap; text-align:left;" id="TBL-15-7-2"  
class="td11"><!--l. 586--><p class="noindent" >The  zero  energy  mode  (equal  rotations)  is  handled  by
adding additional energy term preventing spurious modes.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-15-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-8-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-15-8-2"  
class="td11"><!--l. 588--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-15-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-9-1"  
class="td11">CS properties </td> <td  style="white-space:wrap; text-align:left;" id="TBL-15-9-2"  
class="td11"><!--l. 589--><p class="noindent" >Cross section thickness is required.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-15-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-10-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-15-10-2"  
class="td11"><!--l. 590--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-15-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-11-1"  
class="td11">Nlgeo </td> <td  style="white-space:wrap; text-align:left;" id="TBL-15-11-2"  
class="td11"><!--l. 591--><p class="noindent" >0.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-15-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-15-12-2"  
class="td11"><!--l. 592--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-15-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-13-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-15-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-15-14-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;14: </span><span  
class="content">trplanestressrotallman element summary</span></div><!--tex4ht:label?: x9-2000314 -->
</div><hr class="endfloat" />
</div>
</div>
<div class="center" 
>
<!--l. 595--><p class="noindent" >
<div class="table">
<!--l. 595--><p class="noindent" ><a 
 id="x9-2000415"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-16" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-16-1g"><col 
id="TBL-16-1"></colgroup><colgroup id="TBL-16-2g"><col 
id="TBL-16-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-16-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-16-1-2"  
class="td11"><!--l. 595--><p class="noindent" ><span 
class="cmbx-10">trplanestressrotallman3d</span>                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-16-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-16-2-2"  
class="td11"><!--l. 595--><p class="noindent" >2D linear triangular plane stress element with rotational
DOFs                                                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-16-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-16-3-2"  
class="td11"><!--l. 595--><p class="noindent" >                                                 </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-16-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-16-4-2"  
class="td11"><!--l. 596--><p class="noindent" >Six dofs (D_u, D_v, D_w, R_x, R_y, R_z) are required in
each node.                                                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-16-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-16-5-2"  
class="td11"><!--l. 597--><p class="noindent" >Linear approximation of geometry, quadratic interpolation
of displacements.                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-16-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-16-6-2"  
class="td11"><!--l. 598--><p class="noindent" >Integration   of   membrane   strain   terms   using   gauss
integration formula in 4 points.                                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-16-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-7-1"  
class="td11">Zero energy mode  </td><td  style="white-space:wrap; text-align:left;" id="TBL-16-7-2"  
class="td11"><!--l. 599--><p class="noindent" >The  zero  energy  mode  (equal  rotations)  is  handled  by
adding additional energy term preventing spurious modes.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-16-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-8-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-16-8-2"  
class="td11"><!--l. 601--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-16-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-9-1"  
class="td11">CS properties </td> <td  style="white-space:wrap; text-align:left;" id="TBL-16-9-2"  
class="td11"><!--l. 602--><p class="noindent" >Cross section thickness is required.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-16-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-10-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-16-10-2"  
class="td11"><!--l. 603--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-16-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-11-1"  
class="td11">Nlgeo </td> <td  style="white-space:wrap; text-align:left;" id="TBL-16-11-2"  
class="td11"><!--l. 604--><p class="noindent" >0.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-16-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-16-12-2"  
class="td11"><!--l. 605--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-16-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-13-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-16-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-16-14-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;15: </span><span  
class="content">trplanestressrotallman3d element summary</span></div><!--tex4ht:label?: x9-2000415 -->
</div><hr class="endfloat" />
</div>
</div>
                                                                                           
                                                                                           
                                                                                           
                                                                                           
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